We are tasked with finding the derivative of the function:

$f(x) = x^3 \left( 1 - \frac{4}{x+7} \right)$

Step 1: Apply the Product Rule

The product rule for derivatives states that for two functions $u(x)$ and $v(x)$, the derivative of their product is given by:

$(fg)' = f'g + fg'$

In this case, let:

• $u(x) = x^3$
• $v(x) = \left(1 - \frac{4}{x+7}\right)$

Step 2: Derivatives of $u(x)$ and $v(x)$

1. Derivative of $u(x) = x^3$:

$u'(x) = 3x^2$

2. Derivative of $v(x) = 1 - \frac{4}{x+7}$:

We need to find the derivative of $v(x) = 1 - \frac{4}{x+7}$. The derivative of a constant (1) is 0, and for the fraction:

$\frac{d}{dx} \left( \frac{4}{x+7} \right) = -\frac{4}{(x+7)^2}$

Thus:

$v'(x) = - \left( -\frac{4}{(x+7)^2} \right) = \frac{4}{(x+7)^2}$

Step 3: Apply the Product Rule

Now, using the product rule:

$f'(x) = u'(x) v(x) + u(x) v'(x)$

Substitute the values of $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$:

$f'(x) = 3x^2 \left(1 - \frac{4}{x+7}\right) + x^3 \cdot \frac{4}{(x+7)^2}$

Step 4: Simplify the Expression

Now we will simplify the derivative:

$f'(x) = 3x^2 \left(1 - \frac{4}{x+7}\right) + \frac{4x^3}{(x+7)^2}$

We can leave the derivative in this simplified form.

Would you like further details on how to proceed with simplification or understanding specific parts?

Here are 5 questions related to this derivative:

1. How does the product rule help in differentiating the given function?
2. What are the steps for differentiating rational functions?
3. How do you simplify the result after applying the product rule?
4. Can this expression be factored or further simplified?
5. How does chain rule apply to more complex functions like this?

Tip: When differentiating products of functions, always break them down using the product rule first before simplifying the result!